#### Vol. 11, No. 7, 2018

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Quantitative stability of the free boundary in the obstacle problem

### Sylvia Serfaty and Joaquim Serra

Vol. 11 (2018), No. 7, 1803–1839
##### Abstract

We prove some detailed quantitative stability results for the contact set and the solution of the classical obstacle problem in ${ℝ}^{n}$ ($n\ge 2$) under perturbations of the obstacle function, which is also equivalent to studying the variation of the equilibrium measure in classical potential theory under a perturbation of the external field.

To do so, working in the setting of the whole space, we examine the evolution of the free boundary ${\Gamma }^{t}$ corresponding to the boundary of the contact set for a family of obstacle functions ${h}^{t}$. Assuming that $h={h}^{t}\left(x\right)=h\left(t,x\right)$ is ${C}^{k+1,\alpha }$ in $\left[-1,1\right]×{ℝ}^{n}$ and that the initial free boundary ${\Gamma }^{0}$ is regular, we prove that ${\Gamma }^{t}$ is twice differentiable in $t$ in a small neighborhood of $t=0$. Moreover, we show that the “normal velocity” and the “normal acceleration” of ${\Gamma }^{t}$ are respectively ${C}^{k-1,\alpha }$ and ${C}^{k-2,\alpha }$ scalar fields on ${\Gamma }^{t}$. This is accomplished by deriving equations for this velocity and acceleration and studying the regularity of their solutions via single- and double-layer estimates from potential theory.

##### Keywords
obstacle problem, contact set, coincidence set, stability, equilibrium measure, potential theory
##### Mathematical Subject Classification 2010
Primary: 35R35, 31B35, 49K99